Bohnenblust--Hille and Hardy--Littlewood inequalities by blocks

Abstract

This paper has at least two clear motivations: an aesthetic and a practical. The aesthetic motivation unifies in a single and crystal clear formulation a huge family of inequalities that have been produced separately in the last 90 years in different contexts. From Littlewood's 4/3 inequalities up to today, essentially all inequalities of this kind are encompassed by our main result. But we do not just join inequalities; we also create a family of inequalities overlooked by previous approaches. The practical motivation is to show that this new result (in particular the new inequalities that it contains), besides its own interest, has strength to attack various problems. A first application was already done recently by Maia et al., that, by using our main theorem, substantially improved an inequality of Carando et al. which was impossible to be got by their original method. We also show some more applications.